The decomposition of the hypermetric cone into L-domains

The hypermetric cone HYPn+1 is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone HYPn+1 is polyhedral; one way of seeing this is that modulo image by the covariance map HYPn+1 is the union of a finite set of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into Ldomains. In particular, it is proved that the cone HYPn+1 of hypermetrics on n + 1 points contains exactly 1 2 n! principal L-domains. We give a detailed description of the decomposition of HYPn+1 for n = 2, 3, 4 and a computer result for n = 5 (see Table 2). Remarkable properties of the root system D4 are key for the decomposition of HYP5.